An Alternative Rationale for the Necessity of an Inheritance Tax

Munich Personal RePEc Archive

An Alternative Rationale for the

Necessity of an Inheritance Tax

Harashima, Taiji

Kanazawa Seiryo University

2 May 2020

Online at

https://mpra.ub.uni-muenchen.de/100015/

An Alternative Rationale for the Necessity of an Inheritance Tax

HARASHIMA Taiji*

May 2020

Abstract

In this paper, I present a rationale for an inheritance tax from the point of view of economic rents derived from ranking value and preference in the state of simultaneous optimality of all heterogeneous households. Because there are family lines that have different probabilities of persistently obtaining rents, these rents should be sufficiently taxed to achieve optimality of all heterogeneous households. Income taxes alone cannot completely remove the negative effects of these rents on optimality because it is difficult to distinguish between types of income. Therefore, an inheritance tax is necessary to complement income taxes. An important point is that the inheritance tax is not a tax on capital income—it is a tax on rents. Because sustainable heterogeneity indicates optimality in a heterogeneous population, this type of inheritance tax does not distort the economy. Rather, unless the negative effects of these rents are sufficiently removed by an inheritance tax, the economy is distorted and most households cannot achieve optimality.

JEL Classification code: D63, H21, H24

Keywords: Economic rent; Income tax; Inheritance tax; Inequality; Ranking value and preference; Sustainable heterogeneity

*Correspondence: HARASHIMA Taiji, Kanazawa Seiryo University, 10-1 Goshomachi-Ushi, Kanazawa, Ishikawa, 920-8620, Japan.

1 INTRODUCTION

The justification for inheritance taxes has been studied mostly in the framework of the theory of optimal taxation. However, the conclusions of these studies are mixed and depend greatly on the assumptions used, for example, assumptions on the finiteness of life, borrowing constraints, parental behavior, government behavior, the linearity of taxation, and heterogeneity in a variety of factors. Therefore, the justifiability of an inheritance tax remains an open question.

It has been argued that, in general, an inheritance tax should not be levied because taxes on capital incomes distort the economy and inheritance taxes can be interpreted to be a kind of tax on capital incomes (Atkinson and Stiglitz, 1976; Chamley, 1986; Judd, 1985; Atkeson et al., 1999). However, an inheritance tax (or more broadly a tax on capital incomes) can be justified if we introduce certain assumptions, for example, that there are borrowing constraints or uninsured idiosyncratic heterogeneity (Aiyagari, 1995; Conesa et al., 2009), that parents behave according to particular “bequest technology” (Cremer et al., 2003), that an inheritance tax is only levied on bequests above a certain threshold (Saez, 2013), or that a government maximizes its social welfare function, which consists of general social marginal welfare weights for a population that has heterogeneous bequest tastes and labor productivities (Piketty and Saez, 2013). These assumptions may be economically important, but they may be introduced ad hoc. In any case, there is no current consensus on the rationale for an inheritance tax.

All of these studies, however, are confined within the traditional framework of the theory of optimal taxation. In this paper, I examine the rationale for an inheritance tax from a different perspective, specifically, from the point of view of sustainable heterogeneity (SH) shown in Harashima (2012, 2014, 2017b)1 and rent incomes derived from ranking value and preference shown in Harashima (2017a, 2018a, 2018b, 2018c).2

SH is defined as the state at which all the optimality conditions of all heterogeneous households are satisfied indefinitely. SH can be achieved with appropriate government interventions even if households are heterogeneous in preferences and behave unilaterally. Conversely, unless a government appropriately intervenes, it is difficult to achieve SH. However, if households are heterogeneous only in their preferences, government interventions for SH do not necessarily necessitate an inheritance tax. Here I show that an inheritance tax is needed if households are heterogeneous in their abilities to persistently obtain rent incomes. Harashima (2019a) showed that, if these abilities are heterogeneous across households, the government has

1 _{Harashima (2017b) is also available in English as Harashima (2010).} 2 _{Harashima (2018b) is also available in English as Harashima (2016).}

to intervene with regard to rent incomes to achieve SH. Furthermore, Harashima (2017a, 2018c) showed that rent incomes originating from ranking value and preference exist widely in the economy, but they are unevenly distributed among households. Because of these unevenly distributed rents, the government needs to introduce an inheritance tax to achieve SH. An important point is that an inheritance tax achieves SH, and it is not a tax on capital incomes but on rents. Therefore, it does not distort the economy.

The paper is structured as follows. In Section 2, I discuss ranking value and preference and their relation to persistent rent incomes. I then explore SH and the necessity for government intervention (i.e., an income tax) in Section 3. In Section 4, I develop the argument for the necessity of an inheritance tax and discuss types of inheritance taxes. Finally, concluding remarks are offered in Section 5.

2 PERSISTENT RENT INCOMES

2.1 Ranking value and preference and rent incomes

2.1.1 Ranking value and preference

In this section, the concept of ranking value and preference is explained in briefon the basis of the work of Harashima (2017a, 2018a, 2018b, 2018c).3

2.1.1.1 Ranking value

Value is regarded as reflecting something useful. People feel, obtain, or consume value when using, enjoying, or consuming goods and services. Values derived from practical use have usually been considered in economics, but people also consume values derived from ranking. For example, if a curio is evaluated to be the best among a set of similar items, its price will become very high relative to those of the others, regardless of whether it is practically useful. Its price is so high only because it is the top-ranked item in the group. In this sense, people obtain utility not only from practical uses but also from a sense of ranking.

Therefore, value has two components: practical value and ranking value. Practical value is the value that people feel when consuming a good or service for practical purposes. Ranking value is the value that people feel from the rank of a good or service within a set of similar types of goods or services that people use, possess, or observe (e.g., the ranking of a book in a best-seller list or that of a professional baseball team in a league).

2.1.1.2 Ranking preference

Suppose that goods and services have the following properties: quantity, quality, and rank. Quality is related to practical value, rank is related to ranking value, and quantity is related to both values. Suppose also that the quality and rank of each good or service are given exogenously and fixed. Here, for simplicity, I assume that there is only one type of good or service in the economy, and that all goods or services belong to this type (these goods

or services are hereafter called “goods” for simplicity) and are substitutable for households’ practical uses. Although the goods are substitutable from the point of view of practical uses, they are differentiated from the point of view of rank.

As first developed in Harashima (2018b), let R (= 1, 2, 3, …) be the rank of the

goods. The good with rank R = 1 is most preferred by households, rank R = 2 is the next most preferred, and so on. For simplicity, it is assumed that there is no tied rank. A household’s utility derived from consuming goods with rank R is

𝑢(𝑞𝑛,𝑅, 𝑞𝑙,𝑅, 𝑅) ,

where qn,R and ql,R are the quantity and quality of the good with rank R, respectively. For simplicity, the utility of the household is modified to

𝑢(𝑞̃𝑅, 𝑅) ,

whereq~R is the “quality-adjusted quantity” of the good with rank R, andqR qn,Rql,R

~ _.

The utility function has the following conventional characteristics:

𝜕𝑢(𝑞̃𝑅, 𝑅) 𝜕𝑞̃𝑅 > 0 and 𝜕2_𝑢(𝑞̃ 𝑅, 𝑅) 𝜕𝑞̃𝑅2 < 0 . In addition, for anyrR ,

𝑢(𝑞̃𝑟, 𝑟 + 1) < 𝑢(𝑞̃𝑟, 𝑟) and

𝑢(𝑞̃𝑟, 𝑟 + 2) − 𝑢(𝑞̃𝑟, 𝑟 + 1) > 𝑢(𝑞̃𝑟, 𝑟 + 1) − 𝑢(𝑞̃𝑟, 𝑟) .

2.1.2 Ranking value and superstars

Ranking value and preference confer monopoly powers and thereby rents (which I call

“ranking monopoly rents”) to the producers of high-ranking products because selling ranking values to consumers requires no additional cost; that is, the marginal cost of producing a ranking value is zero. Therefore, producers of highly ranked products can set

prices above their marginal costs (see Harashima, 2017, 2018b). If households’ ranking

preference is strong enough, therefore, the highest-rank producer (e.g., the best player in a professional sport) can be a superstar. Harashima (2018a) showed that individual players in team sports can be also superstars and that the origin of the extremely high compensation of superstars in team sports is also ranking value and preference.

2.1.3 Product differentiation and monopoly rents

The importance of product differentiation has been emphasized in the study of business administration. The differentiation strategy is one of the three fundamental strategies in

Porter’s generic strategies (Porter, 1980, 1985). Harashima (2017a) showed that product differentiation is important because it provides firms monopoly powers, profits, and rents that originate from household ranking preferences. Because the strategy of product differentiation is so important (Porter, 1980, 1985) and is actually pursued by many companies, the monopoly rents generated from differentiation will be large and widespread across the economy today and in the future.

Who should receive the ranking monopoly rents that are derived from product differentiation—shareholders, ordinary employees, consumers, or executives? It seems reasonable that they should be distributed according to the level of contribution to the generation of the ranking monopoly rents, but determining the various levels of contribution by the different parties is difficult.

The monopoly rents of firms that successfully differentiate their products commonly originate in ranking value and preference in much the same way as occurs in professional sports. In addition, teams and firms share the common feature that, to obtain a higher rank, it is essential to hire more talented employees—in particular, players or executives. In these respects, a team and a firm can be seen as the same kind of economic agents or organizations in that they are the beneficiaries of households’ ranking preferences. Therefore, it is highly likely that the mechanism of executive compensation

is the same as that for players’ salaries in team sports and that some executives will be compensated like superstars in team sports.

This does not mean that none of the ranking monopoly rent is distributed to shareholders, ordinary employees, or consumers. Indeed, it is quite likely that some of these rents are also distributed, for example, to shareholders in the form of share dividends and capital gains.

2.2 Persistent rent incomes

2.2.1 Family lines

It is likely that there are family lines (FLs) and that some of these will consist of households that are descended from common ancestors and that share similar traits. In addition, in accordance with custom and for other reasons, many groups of people mostly marry within the same or similar groups. Various abilities are exogenously given to people, but the distribution is biased. Therefore, it is reasonable to assume for simplicity that there are different FLs consisting of households that commonly share a given probability of obtaining ranking monopoly rents; moreover, this probability differs from the probabilities of households in other FLs.

Here, I examine ranking monopoly rents in an economy with FLs that are heterogeneous in the probability of obtaining those rents.

2.2.2 Homogeneous FLs

Before examining the case of heterogeneous FLs, I first examine the case of homogeneous FLs for comparison. Suppose that there are two FLs (FL1 and FL2). Each FL equally consists of N(= 1,2,3, …, N) households. A generation of each household lives only one period, but any household continues to exist over many generations indefinitely (i.e., there are equivalently indefinitely living households). Any household in both FLs equally obtains ranking monopoly rents (𝑧_𝑡 = 𝑧 + 𝜖_𝑡) in period t (i.e., during the period of generation t) with probability 1

2𝑁; that is, one and only one of 2N households in the two

FLs on average obtains 𝑧_𝑡 in period t where z is a constant and 𝜖_𝑡 is i.i.d. with mean zero. In other words, any household obtains ranking monopoly rents in every 2N

generations on average. Note that obtaining ranking monopoly rents means that, for example, in a given generation, a member of a household obtains a post as high-ranking executive in a large company and earns a very high level of compensation. If a household cannot obtain ranking monopoly rents in period t (i.e., during the period of generation t), its incomes decreases by 𝑧𝑡

2𝑁−1 in this period (generation) because economic rents

indicate transfers of incomes between households, and those transfers are unrelated to production costs.

the expected 𝑧_𝑡 minus expected 𝑧𝑡

2𝑁−1

)

𝐸(𝑧𝑡) =_{2𝑁 𝑧 − (1 −}1 _2𝑁)1 _{2𝑁 − 1 = 0}𝑧 ,

where E is the expectation operator. That is, the expected net ranking monopoly rents are zero for any household in either FL, which indicates that 𝑧_𝑡 has no persistent effect. Hence, if all households are homogeneous, any ranking monopoly rents are interpreted to be “temporarily” obtained.

2.2.3 Heterogeneous FLs

Next, I examine the case of heterogeneous FLs. In this case, the two FLs are identical except for the probability of obtaining ranking monopoly rents. Each household in FL1

obtains ranking monopoly rents in a period with probability 2

3𝑁 (in other words, it

obtains them in every 3𝑁

2 generations on average). On the other hand, each household

in FL2 obtains them in a period with probability 1

3𝑁 (in other words, it obtains them

once in every 3N generations on average). The probability of obtaining rents in households in FL1 is therefore two times higher than that of households in FL2. On average, one and only one of 2N households in the two FLs obtains 𝑧_𝑡 in each period.

For a household in FL1, the expected ranking monopoly rents are

𝐸(𝑧𝑡) =_{3𝑁 𝑧 − (1 −}2 _3𝑁)2 _{2𝑁 − 1 =}𝑧 _{6𝑁 − 2 > 0}𝑧 , and those for a household in FL2 are

𝐸(𝑧𝑡) =_{3𝑁 𝑧 − (1 −}1 _3𝑁)1 _{2𝑁 − 1 = −}𝑧 _{6𝑁 − 2 < 0}𝑧 .

Therefore, the expected ranking monopoly rents of a household in FL1 are positive but those in FL2 are negative, and unlike the case of a homogeneous population, ranking monopoly rents have persistent effects.

At first glance, ranking monopoly rents may look like temporarily (or “luckily”) obtained incomes because they can only be intermittently obtained by any household. However, they can have persistent effects as shown above and are actually “persistently”

obtained rent incomes when households are heterogeneous in the probability of obtaining ranking monopoly rents.

2.2.4 Definition of persistently obtained ranking monopoly rents

The natures of ranking monopoly rents are different depending on whether FLs are homogeneous or heterogeneous. If FLs are homogeneous, ranking monopoly rents are temporary incomes, whereas they are persistent if FLs are heterogeneous because the

𝐸(𝑧𝑡) of households in at least one FL must be positive if FLs are heterogeneous. The heterogeneity in abilities across FLs ultimately generates the persistent nature of ranking monopoly rents.

Consequently, we can define the “persistence” of ranking monopoly rents by whether or not the FLs are heterogeneous in their abilities. In this paper, ranking monopoly rents are considered to be persistent if FLs are heterogeneous in their abilities. The rents are temporary if the FLs are homogeneous in their abilities.

2.3 Temporary rents and other forms of persistent rents

Persistently obtained rent incomes are not limited to persistently obtained ranking monopoly rents. For example, there are rents obtained from natural monopolies or from natural resources like land and oil. I call these various kinds of persistently obtained rent

incomes as a whole “persistent rent incomes.”

Furthermore, the definition of persistent rent incomes given in Section 2.2.4 indicates that there are also temporary rent incomes. If the probability of obtaining any rent income is identical in all heterogeneous households, it is considered to be temporary rent income. There are many examples of temporary tent incomes, including lottery prizes and other one-time “lucky” occurrences that can affect profits and incomes.

Persistent rent incomes can be purchased by and exchanged with other households. Furthermore, a household can exchange its temporary rent incomes for various forms of persistent rent incomes; for example, they can buy shares of companies or land (some parts of share dividends and capital gains from shares are types of persistently obtained ranking monopoly rents). An owner of temporary rent incomes therefore can become an owner of persistent rent incomes.

3 SUSTAINABLE HETEROGENEITY AND

INCOME TAX

In this section, I explain SH briefly on the basis of the work of Harashima (2012, 2014, 2017b, 2019a).4

3.1.1 The SH model

Suppose for simplicity that there are only two economies―Economy 1 and Economy 2—

that are identical except for the rate of time preference (RTP). Each economy consists of its own identical households, respectively. Let θ1 and θ2 be the RTPs of households in

Economies 1 and 2, respectively, and θ1 < θ2. The population growth rate is zero in both

economies. The two economies are fully open to each other, and goods, services, and capital are freely transacted between them, but labor is immobilized in each economy. Because the economies are fully open, they are integrated through trade and form a combined economy. The combined economy can be interpreted as the world economy (the international interpretation) or the national economy (the national interpretation). Usually, the concept of the balance of payments is used only for international transactions, but because both national and international interpretations are possible, this concept and terminology are also used for the national economy model in this paper.

Because a balanced growth path requires Harrod-neutral technological progress, the production function of Economy i is assumed to be

𝑦𝑖,𝑡 = 𝐴𝑡𝛼𝑘𝑖,𝑡1−𝛼

for i = 1 or 2, where yi,t and ki,t are the per capita production and capital, respectively, of

Economy i in period t; At is the technology in period t; and α ( 0 < α < 1) is a constant. The current account balance in Economy 1 is τt and that in Economy 2 is –τt. The accumulated current account balance

∫ 𝜏𝑠𝑑𝑠 𝑡 0

mirrors capital flows between the two economies, and the economy with current account

surpluses invests them in the other economy. Since 𝜕𝑦1,𝑡

𝜕𝑘1,𝑡 (= 𝜕𝑦_2,𝑡 𝜕𝑘_2,𝑡) are returns on investments, 𝜕𝑦1,𝑡 𝜕𝑘1,𝑡∫ 𝜏𝑠𝑑𝑠 𝑡 0 and 𝜕𝑦2,𝑡 𝜕𝑘2,𝑡∫ 𝜏𝑠𝑑𝑠 𝑡 0

represent income receipts or payments on the assets that an economy owns in the other economy. Hence, 𝜏𝑡−𝜕𝑦_𝜕𝑘2,𝑡 2,𝑡∫ 𝜏𝑠𝑑𝑠 𝑡 0

is the balance on goods and services of Economy 1, and

𝜕𝑦1,𝑡

𝜕𝑘1,𝑡∫ 𝜏𝑠𝑑𝑠 𝑡

0 − 𝜏𝑡

is that of Economy 2. Because the current account balance mirrors capital flows between the economies, the balance is a function of capital in both economies, such that

𝜏𝑡= 𝜅(𝑘1,𝑡, 𝑘2,𝑡) .

The government (or an international supranational organization under the international interpretation) can intervene in the activities of Economies 1 and 2 by transferring money between the two economies. The amount of transfer from Economy 1 to Economy 2 in period t is gt, and it is assumed that gt depends on capital, such that

g_𝑡 =g̅_𝑡𝑘1,𝑡 .

g

̅_𝑡 is an exogenous variable for households and firms and is appropriately adjusted by the government (or an international supranational organization) in every period so as to achieve SH. Because 𝑘_1,𝑡= 𝑘_2,𝑡 and 𝑘̇_1,𝑡 = 𝑘̇_2,𝑡,

g_𝑡 =g̅_𝑡𝑘1,𝑡= g̅_𝑡𝑘2,𝑡 .

Each household in Economy 1 maximizes its expected utility

𝐸 ∫ 𝑢1(𝑐1,𝑡)exp ∞

0 (−𝜃1𝑡)𝑑𝑡 subject to

𝑑𝑘1,𝑡

𝑑𝑡 = 𝐴𝛼𝑘1,𝑡1−𝛼− 𝑐1,𝑡+ (1 − 𝛼)𝐴𝛼𝑘1,𝑡−𝛼(∫ 𝜏𝑠

𝑡

0 𝑑𝑠 + 𝑧0) − 𝜏𝑡−g̅𝑡𝑘1,𝑡 , and each household in Economy 2 maximizes its expected utility

𝐸 ∫ 𝑢2(𝑐2,𝑡)exp ∞ 0 (−𝜃2𝑡)𝑑𝑡 subject to 𝑑𝑘_{𝑑𝑡 = 𝐴}2,𝑡 𝛼_𝑘 2,𝑡 1−𝛼 _{− 𝑐} 2,𝑡− (1 − 𝛼)𝐴𝛼𝑘2,𝑡−𝛼(∫ 𝜏𝑠 𝑡 0 𝑑𝑠 + 𝑧0) + 𝜏𝑡+g̅𝑡𝑘2,𝑡 ,

where ci,t is the per capita consumption of Economy i in period t, ui is the utility function of Economy i, and E is the expectation operator.

3.1.2 SH

Harashima (2017b) showed in an endogenous growth framework that, if and only if

lim

𝑡→∞

𝑐̇_1,𝑡

𝑐_1,𝑡 = 𝑡→∞lim

𝑐̇_2,𝑡

𝑐_2,𝑡 = constant, all the optimality conditions of both economies are

satisfied (i.e., SH is achieved). Even if the government does not intervene (i.e., g̅_𝑡 = 0), SH is achieved if the economies behave multilaterally in the sense that each economy behaves fully considering the optimality conditions of the other economy. On the other hand, if the economies behave unilaterally in the sense that an economy behaves without considering the other economy’s optimality conditions, SH is not achieved unless a government appropriately intervenes. The reason why SH can be achieved in the cases of multilateral behaviors and appropriate government intervention is that the capital accumulation of the more advantaged Economy 1 is restrained by the multilateral behavior and appropriate government intervention. If SH is achieved, the growth rates of consumption in both economies are equal:

lim 𝑡→∞ 𝑐̇1,𝑡 𝑐1,𝑡 = lim𝑡→∞ 𝑐̇2,𝑡 𝑐2,𝑡= 𝜀 −1_[(𝜛𝛼 𝑚v) 𝛼 (1 − 𝛼)−𝛼₋𝜃1 + 𝜃2 2 ] , (1)

where m, v, and 𝜛 are positive constants, and 𝜀 = −𝑐1,𝑡𝑢1

′′

𝑢₁′ = − 𝑐_2,𝑡𝑢₂′′

𝑢₂′ is the degree of

Harashima (2017b) indicated that SH also exists, even in the framework of exogenous growth (e.g., Ramsey-type growth models) with a heterogeneous population. The capital accumulation of the more advantaged Economy 1 is also restrained at SH in exogenous growth models. Hence, the capital (wealth) that a household in Economy 1 owns at SH is not k1 but k1 + Ψ, where Ψ is a negative constant, and the capital that a

household in Economy 2 owns at SH is not k2 but k2–Ψ, where ki,t is identical for any i

through market arbitration (i.e., ki,t = kt for any i).

Note that Harashima (2017b) showed that the two-economy model can be easily extended to multi-economy models, and the results in multi-economy models are basically the same as those in the two-economy model.

3.2 Government intervention

3.2.1 Government intervention when no rent income exists

lim _𝑡→∞g̅_𝑡 =𝜃2− 𝜃1

2 ,

then SH is achieved even if Economy 1 behaves unilaterally, and equation (1) is satisfied.5 When SH is achieved, Economies 1 and 2 consist of a combined economy (Economy

1+2) with twice the population and an RTP of 𝜃1+𝜃2

2 . Suppose that there is a third

economy with RTP of θ3, and it is identical to Economies 1 and 2 except for its RTP.

Because Economy 1+2 has twice the population of Economy 3, if a government intervenes such that the amount of (positive or negative) transfers from a household in Economy 1+2 to households in Economy 3 in period t is g_𝑡 =g̅_𝑡𝑘_3,𝑡=g̅_𝑡𝑘_𝑡,6 SH is achieved where k3,t is capital in Economy 3 in period t, and

lim _𝑡→∞g̅_𝑡= 𝜃3− 𝜃

1+𝜃2

2

3 .

Remember that ki,t is identical for any i through arbitration, as shown in Section 3.1.2. By

iterating similar procedures, if a government’s (positive or negative) transfers from a household in Economy 1+2+ ∙ ∙ ∙ + (H– 1) to households in Economy H are made by

5 _{Households in Economy 2 share the transfers equally with each other.} 6 _{Households in Economy 3 share the transfers equally with each other.}

lim_𝑡→∞̅g_𝑡 =𝜃𝐻−

∑𝐻−1𝑞=1𝜃𝑞

𝐻 − 1

𝐻 ,

then SH is achieved.7

3.2.2 Government intervention when rent incomes exist

Harashima (2019a) showed that SH is achieved even if the government does nothing with regard to rent incomes if they are temporary. If they are persistent, however, SH cannot be achieved unless the government appropriately intervenes. In this section, I briefly explain how a government should intervene for persistent rent incomes on the basis of the work of Harashima (2019a).

Suppose for simplicity that there are only two FLs and there is only one household in each of the two FLs. The households in both FLs are identical except for RTP and the probability of obtaining rent incomes. Only the household in FL1 obtains rent incomes zt in period t. As a result, the incomes of the household in FL2 are reduced by zt in period t. Suppose also that the household in FL1 obtains zt in every period; that is, zt is persistent and that 𝑧_𝑡 = 𝑧̅𝑘_1,𝑡 where 𝑧̅ is a constant. In this case, if the government appropriately controls the value of g̅_𝑡 and

lim

𝑡→∞g̅𝑡= 𝑧̅ +

𝜃2− 𝜃1

2 , (2)

SH is achieved. That is, if rent incomes are persistent, fiscal transfers from the household in FL1 to the one in FL2 by 𝑧_𝑡= 𝑧̅𝑘_1,𝑡 are needed to achieve SH in addition to the transfers for heterogeneous preferences.

3.3 Income tax on persistent rent incomes

Next, I examine how a government should impose income tax on zt based on the two-FL model used in Section 2 in which there are N households and a household in FL1 obtains

zt in a period with probability 2

3𝑁 and that in FL2 obtains zt in a period with probability 1

3𝑁, with zt defined as persistent rent incomes, as defined in Section 2.2.4. Suppose for

simplicity that there is no technological progress and 𝑘_𝑖,𝑡 is constant for any t, and thereby 𝑧_𝑡 = 𝑧 + 𝜖, as was also the case in Section 2.2.

The result in Section 2.2 suggests that one of the easiest ways to impose income taxes on zt is that any household in FL1 pays income taxes 𝑧

6𝑁−2 in any period (i.e.,

even in the periods when it does not obtain zt), and money equivalent to 𝑧

6𝑁−2 is

transferred to each household in FL2 in any period (i.e., even in the periods when it obtains zt). However, this kind of income tax will not be practically feasible because it is difficult for a government to correctly know which FL a household belongs to. In addition, this income tax will not be supported by households in FL1 because income taxes are imposed on them on zt in any period, regardless of whether they actually obtained zt. At the same time, households in FL2 constantly receive transfers in any period regardless of whether they obtained zt.

Another possible way is to impose income taxes on zt only when a household obtains zt. These income taxes on zt are imposed in any period but only on the household that actually obtains zt in each period, regardless of FL. In this case, the tax rate on zt should be 100% for the following reason. Suppose that the tax rate is χ (0 < χ≤ 1) and the amount of income tax is χzt. If 0 < χ < 1,

lim_𝑡→∞g̅_𝑡 = 𝜒𝑧̅ +𝜃2− 𝜃₂ 1 < 𝑧̅ +𝜃2 − 𝜃₂ 1 ,

and therefore equation (2) cannot be satisfied. It is only when 𝜒 = 1 that equation (2) can be satisfied and SH can be achieved.

4 NECESSITY OF AN INHERITANCE TAX

4.1 Difficulty in estimating z

Even if a government decides to impose an income tax on zt, restricting it to only households that obtain zt in each period, another problem arises. Before imposing an income tax on zt, a government has to distinguish between zt and other household income, particularly between persistent and temporary rent incomes. In other words, the government would ideally know the exact and correct value of zt in each period. However, in reality, it is extremely difficult to distinguish between persistent and temporary rent incomes, or in other words, to estimate how much luck contributed to a household’s current income. Correctly distinguishing between them would require observing and intensely analyzing the incomes of all households in all FLs over many generations in detail.

conservative estimate of the value of zt (i.e., a relatively smaller one) because households will be much less satisfied (angrier) if zt and the corresponding income tax are overestimated than if they are underestimated. Therefore, a government will more greatly fear the case of overestimating zt than that of underestimating it.

4.2 Unrealized capital gains

The basic mechanism of achieving SH through government interventions is that the capital accumulation of a household in FL1 is compulsorily restrained and that in a household in FL2 is enhanced by the government. This mechanism implies that taxation implemented to achieve SH should target not only income but also capital owned by households. This aspect is important if there are incomes that cannot be easily identified by the government. Suppose that two households obtain the same amount of incomes that are recognized by the government, but in addition to these incomes, one of the two households also obtains incomes that cannot be identified by governmentand therefore not taxed. In this case, although the amount of income taxes imposed are the same for both households, the capital held by the two households will grow differently, which means that the mechanism of achieving SH cannot work if it only depends on an income tax.

Do uneasily identifiable incomes exist? Unfortunately, they do, for example, as unrealized capital gains. In many countries, capital gains are not taxed until they are realized because, from a practical standpoint, it is difficult to tax unrealized capital gains. Hence, if a household keeps a large amount of capital gains unrealized over generations, it can evade a large amount of income tax in the absence of government intervention. As a result, that household’s capital accumulation cannot be sufficiently controlled by the government to achieve SH.

4.3 Inheritance tax as a necessity to achieve SH

If income taxes are not necessarily sufficient as a tool to achieve SH for the reasons discussed in Sections 4.1 and 4.2, another type of tax that can help achieve SH is needed. One such tax is the inheritance tax.

4.3.1 Potential of an inheritance tax

If a variable and its components fluctuate largely (in this case, income and its component parts), it can be difficult to distinguish between the component elements (in this case, persistent and temporary incomes). Flow variables can fluctuate widely, and income is a flow variable. On the other hand, stock variables generally do not fluctuate as widely, and capital is a stock variable. Hence, the difficulty in distinguishing between the components

in capital will be much lower than it is for income. For example, if the capital owned by a household in a FL persistently deviates upwards from the average over many generations, it is highly likely that that capital is being accumulated largely as a result of persistent rent incomes because a household generally will not have a run of extraordinarily good luck over many generations.

Because the tax base of an inheritance tax is made up of assets that are roughly equivalent to capital, the problem of distinguishing between the temporary and persistent components will be much smaller in the case of inheritance tax than it will be in the case of an income tax. A government will be able to at least roughly correctly estimate accumulated capital resulting from persistent rent incomes by observing a household’s capital holdings. Because of this nature, an inheritance tax has the potential to be a useful supplement to the income tax to achieve SH. The importance of the ability of a government to distinguish between components is particularly important when it sets inheritance tax exemptions, as is discussed in Section 4.6.2.

4.3.2 Necessity of an inheritance tax

Suppose again that there are two FLs and that there is only one household in each FL. In addition, time is discrete and there is no technological progress. For simplicity, assume that any generation of a household in each FL equally lives for only one period, and only the household in FL1 obtains persistent rent incomes zt and that it obtains them in every period (i.e., in every generation). The government underestimates 𝑧̅ and cannot identify part of 𝑧̅ for the reasons discussed in Sections 4.1 and 4.2. Therefore, income taxes are imposed according to

g̅_𝑡 = (𝑧̅ − 𝑧̌) +𝜃2− 𝜃₂ 1 , (3)

where 𝑧̌ is a constant, 𝑧̅ − 𝑧̌ indicates the underestimated amount of rent income, and

0 < 𝑧̌ < 𝑧̅. Equation (3) indicates that the amount of income taxes will be less than the necessary amount for SH because equation (2) must be satisfied to achieve SH. Note that per capita capital used (not owned) by FL1 and FL2 are still 𝑘_1,𝑡 and 𝑘_2,𝑡, respectively, and that 𝑘_1,𝑡 = 𝑘_2,𝑡.8 Because the income tax is imposed following equation (3), however, the household in FL1 owns more capital than 𝑘_1,𝑡, and the household in FL2 owns less capital than 𝑘_2,𝑡.

To achieve SH, therefore, the government introduces an inheritance tax in

8 _{Capitals (equivalently, machines, equipment, and so on) can be used by households (workers) even if}

addition to the income tax. The inheritance tax is imposed on the households in both FLs in every period (equivalently for every generation) such that the amount of inheritance tax for the household in FL1 is

𝛿(𝑘1,𝑡+ 𝑧̌𝑘1,𝑡− 𝑘̅) , and that for FL2 is

𝛿(𝑘2,𝑡− 𝑧̌𝑘2,𝑡− 𝑘̅) ,

where δ (0 < δ ≤ 1) is the constant inheritance tax rate and 𝑘̅ is the constant tax exemption that is common to both FLs. 𝑘_1,𝑡+ 𝑧̌𝑘_1,𝑡 and 𝑘_2,𝑡− 𝑧̌𝑘_2,𝑡 indicate the respective capital (assets) of households in FL1 and FL2 after paying the income tax (𝑧̅ − 𝑧̌)𝑘_1,𝑡 and receiving the transfers with regard to heterogeneous RTPs. The tax exemption is set to satisfy

𝑘2,𝑡− 𝑧̌𝑘2,𝑡 < 𝑘̅ < 𝑘1,𝑡+ 𝑧̌𝑘1,𝑡 .

Because the value of 𝑘_2,𝑡− 𝑧̌𝑘_2,𝑡− 𝑘̅ is negative, the household in FL2 is exempted from the inheritance tax, and it instead receives a transfer equivalent to the amount of the inheritance tax on the household in FL1 from the government. Therefore, because of the inheritance tax, the capital owned by the household in FL1 is reduced from 𝑘_1,𝑡+ 𝑧̌𝑘_1,𝑡 to (1 − 𝛿)(𝑘_1,𝑡+ 𝑧̌𝑘_1,𝑡), but that owned by the household in FL2 is unchanged (i.e.,

𝑘2,𝑡− 𝑧̌𝑘2,𝑡) before receiving the positive government transfers with regard to the inheritance tax.

For simplicity, suppose that the amount of 𝑧̌𝑘_1,𝑡 is constant such that

𝑧̌𝑘1,𝑡= 𝑧̌𝑘̃1 = 𝑧𝑅 =constant (4) for any t where 𝑘̃_1,𝑡 is 𝑘_1,𝑡 at SH in the case that there is no persistent rent income, and that for any t,

𝑘̅ < 𝑘1,𝑡+ 𝑧𝑅 = 𝑘1,𝑡+ 𝑧̌𝑘̃1 . (5) Because time is discrete and there is no technological progress,

𝑘1,𝑡+1 = 𝑘1,𝑡+ 𝑧𝑅 − 𝛿(𝑘1,𝑡+ 𝑧𝑅− 𝑘̅) = (1 − 𝛿)(𝑘1,𝑡+ 𝑧𝑅) + 𝛿𝑘̅ . By iteration, 𝑘1,𝑡+2 = (1 − 𝛿)(𝑘1,𝑡+1+ 𝑧𝑅) + 𝛿𝑘̅ ∙ ∙ 𝑘1,𝑡+𝑚 = (1 − 𝛿)(𝑘1,𝑡+𝑚−1+ 𝑧𝑅) + 𝛿𝑘̅ , and thereby, 𝑘1,𝑡+𝑚 = (1 − 𝛿)𝑚𝑘1,𝑡+ (1 − 𝛿)𝑧𝑅∑(1 − 𝛿)𝑞−1 𝑚 𝑞=1 + 𝛿𝑘̅ ∑(1 − 𝛿)𝑞−1 𝑚 𝑞=1 . Hence, lim 𝑚→∞𝑘1,𝑡+𝑚 = 𝑧𝑅( 1 − 𝛿 𝛿 ) + 𝑘̅ =constant. (6)

Equation (6) indicates that the capital owned by the household in FL1 does not explode but eventually converges at a certain constant value, that is, 𝑧_𝑅(1−𝛿_𝛿 ) + 𝑘̅.

By equations (4) and (6), if the tax rate is set to be

𝛿 = 𝑧̌𝑘̃1

𝑧̌𝑘̃1+ 𝑘̃1− 𝑘̅ , (7) then

lim

𝑚→∞𝑘1,𝑡+𝑚 = 𝑘̃1 . (8) If equations (7) and (8) are satisfied and the revenues from the inheritance tax are appropriately transferred to the household in FL2, SH is achieved. Let the tax rate that satisfies equation (7) (the “SH tax rate”) be 𝛿_𝑆𝐻. By equation (7), as the value of 𝑘̅ approaches the value of 𝑘̃₁, 𝛿_𝑆𝐻 approaches its upper bound (i.e., unity), and if 𝑘̅ = 𝑘̃₁,

𝛿𝑆𝐻 = 1. If 𝑘̅ = 0, 𝛿𝑆𝐻 =_1+𝑧𝑧̌_̌, and if 𝑘̅ > 𝑘̃1, the SH tax rate cannot be set because δ ≤ 1.

deviate upwards largely from that owned by the household in FL2 if the inheritance tax rate is the SH tax rate. With the appropriate tax rate, therefore, an inheritance tax can supplement the income tax to achieve SH. In this sense, it may be seen as the final “safety net” for achieving SH.

An important point is that the state indicated by SH is efficient because all of the optimality conditions of all heterogeneous households are satisfied. In addition, the inheritance tax for achieving SH is not a tax on capital incomes, but rather it is a tax on rents. That is, the inheritance tax does not hinder efficiency. On the contrary, it is a necessity for efficiency because SH cannot be achieved without it.

4.4 Way to set the SH tax rate approximately

Equations (7) and (8) indicate that before setting δ, a government needs to know the value of 𝑧̌. It will, however, be difficult to setδ for similar reasons as those discussed in Section 4.1 for the value of zt in the case of an income tax. If δ is incorrectly set such that 𝛿 <

𝛿𝑆𝐻, increases in capital of the household in FL1 will accelerate, and SH cannot be achieved. If it is incorrectly set in the opposite direction such that 𝛿 > 𝛿_𝑆𝐻, 𝑘_1,𝑡 will decrease to 𝑘̅, which also means that SH cannot be achieved because 𝑘̅ < 𝑘_1,𝑡+ 𝑧_𝑅 for any t as indicated in inequality (5) and thereby 𝑘̅ < 𝑘̃₁. Moreover, as shown in Section 4.3.2, if 𝑘̅ > 𝑘̃₁, the SH tax rate cannot be set. How can a government solve this problem? Harashima (2019b)9 _{showed that, in the case of heterogeneous preferences, if a}

government in a democratic society adjusts the transfers for SH to make the number of votes cast in elections in response to increases in the level of economic inequality equivalent to the number cast in response to decreases, then SH can be approximately achieved. In addition, in the case of heterogeneous abilities of obtaining persistent rent incomes, the value of δ will be able to be set in the same manner. If the value of δ is set in this manner, it will be approximately equal to 𝛿_𝑆𝐻 and SH will be approximately achieved.

4.5 Progressiveness

4.5.1 Does progressiveness matter?

As shown in Section 4.3, if 𝑘̅ = 𝑘̃₁, 𝛿_𝑆𝐻= 1. This implies that the progressiveness in the inheritance tax rate does not matter if 𝑘̅ is set appropriately; that is, it is enough to set the single tax rate, 𝛿_𝑆𝐻 = 1. However, in actuality, 𝑘̅ will not be set equal to 𝑘̃₁ because the problem with distinguishing the persistent and temporary elements of capital, although much smaller than in the case of the income tax, will still exist to some extent.

In particular, if 𝑧_𝑡 is small, it will be difficult to distinguish the effect of persistent rent incomes from that of repeated incidences of good luck. To solve this problem, multiple inheritance tax rates may be necessary.

More importantly, households in some FLs can obtain larger amounts of rent income more often than those in other FLs, as discussed in Section 2.2. If an inheritance tax consists of only a single tax rate and this single rate is applied commonly to all heterogeneous FLs, SH may not be achieved. In practice, inheritance taxes have the property of progressiveness in many countries.

4.5.2 Rationale for progressiveness

A multi-FL model is used to examine the necessity of progressiveness because progressiveness only matters when there are multiple heterogeneous FLs with regard to persistent rent incomes. Suppose that there are M FLs (FL1, FL2, …, FLM) that are identical except for the persistent rent incomes zt that households obtain. M is an even number for a technical reason. For simplicity, assume there is only one household in each FL, and that any generation of household in a FL equally lives only one period. If 𝑖 ≤𝑀₂, the household in FLi obtains persistent rent incomes in every period (i.e., in every generation). The amount of persistent rent income after paying the income tax is

𝑧̌𝑖𝑘̃𝑖 = 𝑧𝑅,𝑖

in every period, and 𝑘̃_𝑖 is 𝑘_𝑖,𝑡 at SH in the case where there is no persistent rent income for any FL. It is assumed for simplicity that 𝑧_𝑅,𝑖 is constant for any 𝑖 ≤𝑀₂, and 𝑧_𝑅,𝑖 >

𝑧𝑅,𝑗 if i < j.10 Conversely, if 𝑗 > 𝑀₂, the incomes of household in FLj decreases by

𝑧̌𝑖𝑘̃𝑀−𝑗+1 in every period before the government transfers the money from the inheritance tax to the household in FLj. Here, suppose for simplicity that 𝑘̃_𝑖 is identical for any i(= 1, 2, …, M) and therefore 𝑧̌_𝑖 > 𝑧̌_𝑗 if i < j.

Because any generation of a household lives only one period, the inheritance tax is imposed in every period. The amount of inheritance tax for household in FLq (ITq) for

q= 1, 2, …, M is

𝐼𝑇𝑞 = 𝛿(𝑘𝑞,𝑡+ 𝑧̌𝑞𝑘𝑞,𝑡− 𝑘̅) .

10 _{If a more general environment is assumed such that there are many households in a FL, 𝑧}

𝑅,𝑖> 𝑧𝑅,𝑗 means that a household in FLi can obtain persistent rent incomes more frequently than once in FLj because households can obtain rent incomes only intermittently.

The tax exemption (𝑘̅) is set to satisfy

𝑘𝑗,𝑡− 𝑧̌𝑖𝑘𝑀−𝑗+1,𝑡 < 𝑘̅ < 𝑘𝑖,𝑡+ 𝑧̌𝑖𝑘𝑖,𝑡

for any 𝑖 (≤𝑀₂) and 𝑗 (>𝑀₂). If ITq is negative, the inheritance tax of a household in FLq is exempted. Therefore, because of this inheritance tax, the capital owned by the household in FL𝑖 (≤ 𝑀₂) is reduced from 𝑘̃_1,𝑡 = 𝑘_1,𝑡+ 𝑧̌_𝑖𝑘_1,𝑡 to (1 − 𝛿)(𝑘_1,𝑡+ 𝑧̌_𝑖𝑘_1,𝑡). On the other hand, capital owned by the household in FL𝑗 (>𝑀₂) is unchanged (i.e.,

𝑘𝑗,𝑡− 𝑧̌𝑖𝑘𝑗,𝑡) before the government transfers the money from the revenue of the inheritance tax to the household in FL𝑗 (>𝑀₂) appropriately to achieve SH. With the transfers, the capital owned by the household in FL𝑗 (>𝑀₂) increases to greater than

𝑘𝑗,𝑡− 𝑧̌𝑖𝑘𝑗,𝑡.

By a similar procedure as shown in Section 4.3, if the tax rate were set to

𝛿 = 𝑧̌𝑖𝑘̃𝑖 𝑧̌𝑖𝑘̃𝑖 + 𝑘̃𝑖 − 𝑘̅ for any 𝑖 (≤𝑀₂), then

lim

𝑚→∞𝑘𝑖,𝑡+𝑚= 𝑘̃𝑖 (9) could be satisfied. However, because 𝑧̌_𝑖 is heterogeneous, there is no value of δ that can satisfy equation (9) for any 𝑖 (≤𝑀₂). Conversely, to make equation (9) hold for any

𝑖 (≤𝑀₂), the inheritance tax rate applied to each FL should be heterogeneous across FLs. Hence, I next examine the case of heterogeneous inheritance tax rates. Let 𝛿_𝑖 be the inheritance tax rate for FL𝑖 (≤𝑀₂). Clearly, if

𝛿𝑖 =_𝑧̌ 𝑧̌𝑖𝑘̃𝑖

𝑖𝑘̃𝑖 + 𝑘̃𝑖 − 𝑘̅ (10) is satisfied for any 𝑖 (≤𝑀₂), equation (9) holds simultaneously for any 𝑖 (≤ 𝑀₂). If the

government at the same time transfers the money from the inheritance tax to the household in FL𝑗 (>𝑀₂) appropriately, SH can be achieved. Note that if 𝑘̅ > 𝑘̃_𝑖 for households in some FL𝑖 (≤𝑀₂), a δi that satisfies equation (10) cannot be set for these FLs because δi≤ 1. Because 𝜕𝛿𝑖 𝜕𝑧̌𝑖 = 𝑘̃𝑖(𝑘̃𝑖 − 𝑘̅) (𝑧̌𝑖𝑘̃𝑖 + 𝑘̃𝑖 − 𝑘̅)2 , if 𝑘̃_𝑖− 𝑘̅ > 0 for any 𝑖 (≤ 𝑀₂), 𝜕𝛿𝑖 𝜕𝑧̌𝑖 > 0 .

Therefore, as the amount of inheritance becomes larger, the inheritance tax rate applied to it should increase to achieve SH. That is, inheritance tax rates should be progressive.

4.5.3 Another rationale

There is another rationale for the need for a progressive inheritance tax. As discussed in Section 4.5.1, because the problem of distinguishing between the persistent and temporary elements of capital still exists to some extent, if the amount of capital owned by a household is not sufficiently larger than the average, it will be difficult to distinguish the effect of persistent rent incomes on capital from the effect of repeated good luck. In this case, a government may be still cautious about setting 𝛿_𝑖 as indicated by equation (10).

However, if the amount of capital of a household is sufficiently greater than average, it will be easier to distinguish between them because good fortune does not generally repeat for a long period (i.e., for many generations). A much larger capital amount than average indicates that it is highly likely that the capital was accumulated as a result of persistent rent incomes. As the amount of capital increases, the probability that it was accumulated owing to persistent rent incomes will also increase. Hence, it seems quite reasonable to set a higher inheritance tax rate for a larger amount of inherited capital and a lower rate for a smaller amount.

An important implication of this rationale is that inheritance tax rates should vary depending not on 𝑧̌_𝑖 but on the amount of capital inherited.

4.6 Actual progressive inheritance tax

In many countries, inheritance tax rates do indeed vary depending on the amount of capital inherited, as discussed in Section 4.5.3 and most likely for the same reasons. This kind of inheritance tax is important as a “next best” inheritance tax while the inheritance tax whose rates vary depending on 𝑧̌_𝑖 would be the “best” inheritance tax if it is feasible, and I examine this “next best” inheritance tax in this section.

4.6.1 Inheritance tax rates that vary depending on the amount of

capital inherited

Suppose there is an inheritance tax, and the rates vary depending on 𝑘_𝑖,𝑡+ 𝑧̌_𝑖𝑘_𝑖,𝑡. The tax rates are T1, T2, and T3, and 0 < T1 < T2 < T3 < 1. The tax exemption is set such that 𝑘̃_𝑖 >

𝑘̅ for any 𝑖 (≤𝑀₂). The environment assumed in this section is basically the same as that in Section 4.5.2, but there are N households in each FL, rather than one. In addition, the amount of persistent rent incomes is commonly z for any household and for any period, where z is a constant, but the probability that a household in FLi obtains z (pi) is different across FLs such that

𝑝𝑖 = 2(𝑀 + 1 − 𝑖)_{𝑁𝑀(𝑀 + 1)} (11) while 𝑁 ∑ 𝑝𝑖 𝑀 𝑖=1 = 1 .

Equation (11) indicates that as i becomes smaller, the probability becomes higher, and

that a household in FLi obtains z in every 𝑁𝑀(𝑀+1)

2(𝑀+1−𝑖) generations on average. Hence,

FL1 has the highest probability among the FLs, and the probability that a household obtains z in FL1 is

𝑝1 = _{𝑁(𝑀 + 1)}2 ,

which means that it obtains z once in 𝑁(𝑀+1)

that it always obtains z exactly once in every 𝑁(𝑀+1)

2 generations.

4.6.2 The nature of this inheritance tax

First, I examine a special case where the inheritance tax rate is only T1 for comparison.

Because the tax rate is only T1,

𝑘1,1 = (𝑘1,0+ 𝑧𝑅− 𝑘̅)(1 − 𝑇1) + 𝑘̅ 𝑘1,2 = (𝑘1,1− 𝑘̅)(1 − 𝑇1) + 𝑘̅ 𝑘1,3 = (𝑘1,2− 𝑘̅)(1 − 𝑇1) + 𝑘̅ . . 𝑘1,𝑡 = (𝑘1,𝑡−1− 𝑘̅)(1 − 𝑇1) + 𝑘̅ , and thereby, 𝑘1,𝑡 = (𝑘1,0+ 𝑧𝑅− 𝑘̅)(1 − 𝑇1)𝑡+ 𝑘̅ . (12) If T1 is set sufficiently high, by equation (12),

𝑘1,𝑣 ≅ 𝑘̅ , (13) where 𝑣 =𝑁(𝑀+1₂ )− 1 because 0 < 1 − 𝑇₁ < 1. Because 𝑘̃_𝑖 > 𝑘̅ for any 𝑖 (≤𝑀₂), the state indicated by equation (13) is inconsistent with SH and means that households in FL1 are eventually made poorer than they are at SH because of the inheritance tax. However, if T1 is set sufficiently low, it will take a long time (many generations) to reach

the state indicated by equation (13). Hence, a household in FL1 can obtain z again after

𝑁(𝑀+1)

2 generations, that is, before reaching the state indicated by equation (13). If T1

is set sufficiently low, therefore, a household in FL1 never reaches the state indicated by equation (13). As a result, a household in FL1 can accumulate a larger amount of 𝑘_1,𝑡 than it would at SH and eventually monopolize all capital in the economy.

Next, I examine the case of multiple tax rates T1, T2, and T3. Suppose that T1, T2,

and T3 are set such that

T2 on 𝑘_𝑖,𝑡 when 𝑘̅_{𝐻𝑖𝑔ℎ} ≥ 𝑘_𝑖,𝑡 > 𝑘̅_{𝑀𝑖𝑑𝑑𝑙𝑒}

T3 on 𝑘_𝑖,𝑡 when 𝑘̅_{𝑀𝑖𝑑𝑑𝑙𝑒} ≥ 𝑘_𝑖,𝑡 > 𝑘̅_𝐿𝑜𝑤

so as to reach the state 𝑘_1,𝑡 = 𝑘̃₁ just after 𝑁(𝑀+1)

2 generations where 𝑘̅𝐻𝑖𝑔ℎ >

𝑘̅𝑀𝑖𝑑𝑑𝑙𝑒 > 𝑘̅𝐿𝑜𝑤. If T1 is set sufficiently high, 𝑘_1,𝑡 decreases rapidly to 𝑘̅_{𝐻𝑖𝑔ℎ} but after

it reaches 𝑘̅_{𝐻𝑖𝑔ℎ}, the tax rate applied is changed downwards to T2, and after it reaches

𝑘̅𝑀𝑖𝑑𝑑𝑙𝑒, the lowest rate T3 is applied to the remaining 𝑘_1,𝑡.

An important advantage of this type of inheritance tax is that a government can

more easily make 𝑘_1,𝑡 land softly on 𝑘̃₁ just after 𝑁(𝑀+1)

2 generations. Because T1

can be set sufficiently high, it will not take a long time (i.e., many generations) for 𝑘_1,𝑡 to reach 𝑘̅_{𝐻𝑖𝑔ℎ}, but because T3 can be set sufficiently low, it will take a long time (many

generations) before reaching 𝑘_1,𝑡 ≅ 𝑘̅_𝐿𝑜𝑤. Hence, before reaching 𝑘_1,𝑡 ≅ 𝑘̅_𝐿𝑜𝑤, the government has enough time to adjust or fine-tune T1 , T2, and T3 as needed even if the

initial values of T1 , T2, and T3 do not necessary exactly cause 𝑘_1,𝑡 = 𝑘̃₁ to be realized

immediately after 𝑁(𝑀+1)

2 generations. Because soft landings are possible, this “next

best” inheritance tax appears to be sufficiently useful.

However, being useful does not mean that this inheritance tax system has no drawbacks. Households in FLi ( ≥ 2) experience periods (generations) of 𝑘_𝑖,𝑡 < 𝑘̃_𝑖 and as i increases, more generations will experience 𝑘_𝑖,𝑡 < 𝑘̃_𝑖 because commonly applied tax rates T1 , T2, and T3 have been set so as to make 𝑘_1,𝑡 = 𝑘̃₁ immediately after 𝑁(𝑀+1)

2

generations. Because the interval of obtaining z between generations of a household in FLi (≥ 2) is longer than that of a household in FL1, 𝑘_𝑖,𝑡 of a household in FLi(≥ 2) will be reduced to 𝑘_𝑖,𝑡 < 𝑘̃_𝑖 before a future generation again obtains z while any generation of a household in FL1 can enjoy 𝑘_1,𝑡 ≥ 𝑘̃₁.

Nevertheless, many households in FLs other than FL1 can receive the transfers with regard to the inheritance tax from the government during some generations. In addition, and more importantly, if 𝑘̅_𝐿𝑜𝑤 is set such that it is not too different from 𝑘̃_𝑖 (but still 𝑘̃_𝑖 > 𝑘̅_𝐿𝑜𝑤), households of FLs other than FL1 can at least keep 𝑘_𝑖,𝑡 = 𝑘̅_𝐿𝑜𝑤 ≅

𝑘̃𝑖. It should be possible for a government to set 𝑘̅𝐿𝑜𝑤 such that it is not too different from 𝑘̃_𝑖 because the indistinguishability problem is less severe in the case of the inheritance tax than it is in the case of the income tax. Therefore, a government will be able to roughly estimate the correct amount of accumulated capital resulting from

persistent rent incomes. As a result, differences in 𝑘_𝑖,𝑡 among households do not accelerate; 𝑘_𝑖,𝑡 will be less than 𝑘̃_𝑖 during some generations, but it will be maintained around 𝑘̃_𝑖 for any i(≥ 2) indefinitely.

There are many other variations of this “next best” inheritance tax. For example, some may argue that a scheme where 𝑘̅_𝐿𝑜𝑤 is set to be the average 𝑘_𝑖,𝑡 of all households and T1 , T2, and T3 are set to reach the state 𝑘_1,𝑡 = 𝑘̅_𝐿𝑜𝑤 immediately after

𝑁(𝑀+1) 2

generations is more favorable. Which variation should be adopted should be decided through the various political processes in democratic countries.

5 CONCLUDING REMARKS

Whether inheritance taxes can be justified economically has remained an open question. Most studies of the rationale for inheritance tax have been confined within the traditional framework of the theory of optimal taxation. In this paper, I examined it from a different perspective—that of rent incomes derived from ranking value and preference at SH (Harashima, 2010, 2012, 2014, 2016, 2017b, 2018a, 2018b).

Because there are many different FLs that have different probabilities of obtaining persistent rent incomes, SH cannot be achieved unless some of the capital accumulation of the more advantaged FLs are compulsorily restrained through government interventions. One such tool is the income tax, but it is difficult for a government to be completely successful relying only on an income tax because a government cannot perfectly distinguish between persistent rent incomes and other types of income. Therefore, another type of tax is needed. In this paper, I showed that, if an inheritance tax with an appropriate tax exemption and tax rates is introduced, the drawbacks of an income tax can be overcome and SH can be achieved. In this sense, an inheritance tax is necessary to achieve SH and may even be considered as the final safety net to achieve it.

Inheritance tax rates should be set according to the probabilities of obtaining persistent rent incomes, but this will be difficult to do in practice because it is still difficult to distinguish between the different elements of capital (e.g., persistent and temporary/lucky). A next best solution is to impose an inheritance tax with rates that vary depending on the amounts of capital (assets) households inherit. Although not perfect, SH can be approximately achieved under this type of inheritance tax system.

An important point is that the inheritance tax is not a tax on capital income—it is a tax on rents. Because SH indicates optimality in a heterogeneous population, this type of inheritance tax does not distort the economy. Rather, unless the negative effects of

these rents are sufficiently removed by an inheritance tax, the economy is distorted and most households cannot achieve optimality.

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